Journal of Physics A Mathematical and General

Two scale factor universality in the spherical model

Paul R. Gerber

Published Jan 1, 1975• 16 citations

Abstract

A universal relation, proposed by Stauffer, Ferer and Wortis, between the scale factors in the scaling functions for the free energy and two-spin correlation function, is shown to be valid for the spherical model with general pair interactions, provided an appropriate definition of the correlation length is adopted. The hypothesis remains valid for the spherical model and numerically for the three-dimensional Ising model, when extended to include the scale factor for the magnetic field.

DOI: 10.1088/0305-4470/8/1/013

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Journal of Physics A Mathematical and General

Two scale factor universality in the spherical model

Paul R. Gerber

Published Jan 1, 1975• 16 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/8/1/013

Related papers

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T

data is discussed.

{T}_{c}

, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid

^{4}\mathrm{He}

, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet

{\mathrm{RbMnF}}_{3}

. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of

{V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}

, where

{q}_{X}

is the coordination number,

{{K}_{C}}^{X}

is the dimensionless inverse critical temperature, and

{n}_{X}

{g}_{X}

, and

{l}_{X}

are the universality scale factors.

{V}_{X}

is an exact invariant for the two-dimensional

S=\frac{1}{2}

Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however,

\frac{{V}_{X}}{{l}_{X}}

is an exact invariant.

{\mathrm{O}}_{2}

, Xe, and $\ensuremath{\beta}$-brass and the "Heisenberg-like" systems RbMn

{\mathrm{F}}_{3}

and EuS to within experimental uncertainties (\ensuremath{\sim} 20%). It provides a sensitive and experimentally convenient indicator of symmetry ("universality") class.

Using an elaborate set of simulational tools and statistically optimized methods of data analysis, we investigate the scaling behavior of the correlation lengths of three-dimensional classical

O(n)

spin models. Considering three-dimensional slabs ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{1}\ifmmode\times\else\texttimes\fi{}R,$ the results over a wide range of n indicate the validity of special scaling relations involving universal amplitude ratios that are analogous to results of conformal field theory for two-dimensional systems. A striking mismatch of the $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ extrapolation of these simulations against analytical calculations is traced back to a breakdown of the identification of this limit with the spherical model.

We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-

S

Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.